Pair correlations of one-dimensional model sets and monstrous covariograms of Rauzy fractals

TL;DR

This paper investigates the pair correlation functions of one-dimensional model sets, revealing complex behaviors in covariograms of Rauzy fractals through renormalisation techniques, advancing understanding of their internal structure.

Contribution

It introduces a novel renormalisation approach to analyze pair correlations and covariograms of Rauzy fractals in one-dimensional model sets.

Findings

Covariograms are simple for intervals but complex for Rauzy fractals.

Renormalisation structures reveal unexpected complex behaviors.

Two concrete examples demonstrate wild covariogram behaviors.

Abstract

The averaged distance structure of one-dimensional regular model sets is determined via their pair correlation functions. The latter lead to covariograms and cross covariograms of the windows, which give continuous functions in internal space. While they are simple tent-shaped, piecewise linear functions for intervals, the typical case for inflation systems leads to convolutions of Rauzy fractals, which are difficult to compute. In the presence of an inflation structure, an alternative path is possible via the exact renormalisation structures of the pair correlation functions. We introduce this approach and derive two concrete examples, which display an unexpectedly complex and wild behaviour.